How to create a rotation in local space that is equivalent to the rotation in world space?
Transforms, as shown, are local transforms, not world transforms.  So yes, you can very easily have two orientations that have the same numbers, but different orientations.
The easiest way to copy a world-space orientation is to use a copy rotation constraint-- either a bone constraint or an object constraint, depending on what you want constrained-- in world-&gt;world space, obviously.  That way, Blender does the math for you.
Constraints won't show up in the transform channel; they use the transform channels as inputs that they modify.  If you want, you can make a constraint &quot;real&quot; by selecting the constrainted object and using the &quot;apply visual transform&quot; operation (from the ctrl a menu for me, in either object or pose mode depending on the type of constraint.)  After applying visual transform, you can delete the constraint, or leave it if you'd prefer.
If you're interested in the pure math associated with this, I would recommend reading (and understanding) https://en.wikipedia.org/wiki/Rotation_matrix .  https://en.wikipedia.org/wiki/Basis_(linear_algebra) may also be useful.